A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a,b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps ϕ: A × A → X (for some Banach space X) with the property...

We first study the behavior of weights on a simply connected nilpotent Lie group G. Then for a subalgebra A of L¹(G) containing the Schwartz algebra 𝓢(G) as a dense subspace, we characterize all closed two-sided ideals of A whose hull reduces to one point which is a character.

Let p,q be positive integers. The groups $U_p\left(ℂ\right)$ and $U_p\left(ℂ\right)\times U_q\left(ℂ\right)$ act on the Heisenberg group $H_{p,q}:=M_{p,q}\left(ℂ\right)\times ℝ$ canonically as groups of automorphisms, where $M_{p,q}\left(ℂ\right)$ is the vector space of all complex p × q matrices. The associated orbit spaces may be identified with ${\Pi }_q\times ℝ$ and ${\Xi }_q\times ℝ$ respectively, ${\Pi }_q$ being the cone of positive semidefinite matrices and ${\Xi }_q$ the Weyl chamber $x\in {ℝ}^q:x₁\ge \cdots \ge x_q\ge 0$. In this paper we compute the associated convolutions on ${\Pi }_q\times ℝ$ and ${\Xi }_q\times ℝ$ explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series of convolution...